1. Field of the Invention
The present invention relates to an improvement in means for and to a novel method of so converting electrical signals into sound as to control with high fidelity the acoustic response of a loudspeaker system. More specifically, the invention relates to a loudspeaker system crossover using passive or active circuit topology.
2. Description of the Prior Art and Definition of Terms
In the present state of the art individual loudspeakers or drivers are not capable of accurately reproducing all audio frequencies that are detectable by the human ear. High fidelity loudspeaker systems have been realized in the prior art, however, by dividing the audio frequency spectrum into two or more frequency bands, and applying each of these portions of the audio spectrum to a separate driver or group of drivers.
For this purpose special electrical filters, called crossover networks, have been provided that allow the different drivers or groups of drivers, each adapted for best response to a particular range or band of frequencies, to be combined in a single system capable of wide frequency coverage. The crossover circuit directs the electrical signals of widely varying frequency to the appropriate driver or group of drivers in a multidriver loudspeaker system.
Crossover network filter topologies, in general, belong to three classifications according to the frequencies passed and rejected, as follows:
(1) Low-pass for woofers, PA1 (2) Band-pass for midranges, and PA1 (3) High-pass for tweeters, PA1 s is the complex frequency variable s=.sigma.+jw K and T are real, positive constants; and e=2.718 or, EQU G(s)=Ke.sup.-sT ( 1) PA1 s, K, T, and e defined as above, PA1 (1.1) The woofer and tweeter do not have ideal amplitude and/or phase characteristics. PA1 (1.2) The loudspeaker drivers function in three-dimensional acoustic space in which the simple energy relation of Equation (5) is not valid at all points. PA1 (1.3) The gradual crossover slope (6 dB/octave) allows too much bass energy to enter the tweeter, and too much treble energy to enter the woofer, causing distortion. PA1 (1.4) Greatly minimize the undesirable effects of non-ideal driver amplitude and phase response on total system performance. PA1 (1.5) Minimize the acoustic wave interference between drivers at the crossover frequencies. PA1 (1.6) Reduce total system harmonic and intermodulation distortion. PA1 (1.7) The total system performance must relate to an "acoustic" sum of driver energies in three-dimensional space. PA1 (1.8) The ideal transfer function of Equation (1) is approached using embodiments of the present invention which operate by acoustically summing two or more approximations to ideal "brick-wall" amplitude functions, all having separate mutually-exclusive frequency passbands. If these amplitude functions are carefully chosen, the acoustic sum will not only approach a flat amplitude vs. frequency characteristic, but will also approach a linear-phase vs. frequency characteristic, with, at most, an ambiguity of phase of .+-.2n .pi. radians (where n=0, 1, 2, 3, . . . ) at the crossover frequencies. Observe that if n=0, the drivers are absolutely as well as relatively in phase, and no phase ambiguity exists.
where woofers are low frequency drivers and respond to the low frequencies, midrange drivers respond to the midrange frequencies, and tweeters are high frequency drivers and respond to the high frequencies. Where more than one filter is used, the frequency common to adjacent ranges or passbands is called the crossover frequency.
For "perfect" fidelity it can be demonstrated mathematically that a loudspeaker system crossover using passive or active circuit topology must realize perfectly the ideal all pass transfer function of Equation (1): EQU f(s)=Ke.sup.-sT ( 1)
Where
with
where whichever form the transfer function implied by Equation (1) is relevant to a particular case will become clear when the separate meanings of f(s) and G(s) are defined hereinafter. It is not possible, however, using current technology, to perfectly realize the ideal all pass transfer function in three-dimensional acoustic space with any known loudspeaker system. Accordingly, all real loudspeaker system configurations or designs are based upon an approximation to one or both forms of the ideal transfer function of Equation (1) in three-dimensional space.
The simplest and commonest prior art approximation to the ideal loudspeaker system is based upon a "two-way" design using an assumed ideal woofer and tweeter combined with a simple 6 dB/octave minimum-phase, low-pass-high-pass, cross-over network, as illustrated in FIG. 1. Mathematically, this approach takes the ideal transfer function of Equation (1) and attempts to reduce it to a function that is independent of frequency, ideally a constant. This is accomplished by expanding the ideal transfer function of Equation (1) in a power series, as follows: when K=1, then ##EQU1## Taking only the first term of this series of Equation (2) gives: ##EQU2## Those skilled in the art will recognize Equation (3) as the simple one-pole low pass transfer function.
If the term s in Equation (3) is replaced by (1/sT).sup.2, the complementary high-pass transfer function with cross-over frequency 1/T is obtained: ##EQU3## Plots of the complementary amplitude response f.sub.1 (s) and f.sub.2 (s) of Equations (3) and (4) are given in FIG. 2.
Equation (5) below shows that the sum of the simple low-pass function of Equation (3) and its complementary high-pass function of Equation (4) is unity, that is, a constant that is independent of frequency: ##EQU4##
If an ideal woofer were connected to a cross-over network having the transfer response of Equation (3) and an ideal tweeter were connected to a crossover network having the transfer response of Equation (4), and the woofer and tweeter were combined in a single system, the result would be a "perfect" loudspeaker system. Its amplitude response would be perfectly flat for all frequencies and there would be no phase shift at any frequency.
Serious problems arise, however, when it is attempted to construct a practical loudspeaker system following the foregoing design procedure. These problems arise from three distinct causes:
Even with the reservations just mentioned, the simple two-way loudspeaker system of FIG. 2 approaches the ideal to a degree sufficient to achieve moderately satsifactory performance.
It should be noted that the transfer function f(s) as discussed here so far is that of the electrical cross-over circuits alone, i.e., f(s) is defined as: ##EQU5## which, in words, represents the ratio of electrical energy at the output of a general crossover filter circuit, or combination of filter circuits including a complete crossover system, to the electrical energy at the input, expressed as a function of complex frequency.
The assumption has been implicitly made so far in this discussion that the transfer function of the loudspeaker drivers (defined immediately hereinafter) is either unity, or an "ideal delay" and thus may be ignored. This is, at best, only approximately true. The transfer function of a loudspeaker driver is an electroacoustical quantity and may be defined as the ratio of the sound pressure at a point in the listening environment to the electrical energy input to the driver terminals; this expresion being a ratio of terms in complex frequency: ##EQU6##
Thus one can consider the overall transfer function of a complete speaker system taken as a whole--crossover plus loudspeaker drivers--which would be the product of the two abovementioned transfer functions and would be defined as: ##EQU7## The transfer function G(s) represents the ratio of acoustic sound pressure at a particular point in the listening environment to the electrical energy applied to the input terminals of the speaker system, both as a function of complex frequency. The terms P.sub.2 (s) and Q.sub.2 (s) will contain the poles and zeros (p-z) of the loudspeaker drivers as well as the p-z of the crossover elements.
Well-designed loudspeaker drivers possess a band of frequencies in which the amplitude response is flat to desired accuracy, and phase response is linear to desired accuracy--such loudspeaker drivers may be referred to as "ideal" and will possess an electroacoustical transfer function H(s) which can be considered to be unity or a constant. Loudspeaker drivers will be considered to possess ideal delay--i.e., to approximate Equation (1) to a high degree of accuracy within their respective frequency bands of best performance--unless specifically stated otherwise. Hereinafter, transfer functions designated as f(s) will always relate to the crossover only, while transfer functions designated G(s) will always relate to the crossover filter circuits plus the loudspeaker drivers, with definitions as set forth hereinbefore.
Those skilled in the art will understand that the units of the right-hand terms contained in Equation (1a) are voltages and currents which relate to the electrical circuit of the crossover network under consideration. The units of the right-hand terms of Equations (1b) and (1c) are voltages, currents, forces, velocities, pressures, and volume velocities, which relate to the electrical circuit (crossover), the mechanical circuit (the loudspeaker driver), and the acoustical circuit (the air surrounding the loudspeaker driver). For a numerical solution to specific forms of Equations (1b) and (1c), or any other equation contained herein with mixed electrical, mechanical, and/or acoustical circuits, such as Equation (8) expressed hereinafter, the method of "dynamic analogies" must be employed. This method is amply treated in the text Acoustics, Chapters 3 and 5, by Leo Beranek, McGraw-Hill, 1947. Hereinafter, this method of dynamic analogies will be implicitly assumed to have been applied whenever an "acoustic" sum is discussed with respect to an equation containing "mixed", i.e., electroacoustical units.
Returning to the earlier discussion, it is observed that closer approximations to the ideal transfer function of Equation (1) have traditionally been realized by taking more terms of the infinite series of Equation (2), again attempting to reduce the result to a function independent of frequency, and then using such function as a basis for design considerations. Some of these methods have been treated in the prior art and in particular, in several issues of the Audio Engineering Society Journal, specifically in the following articles: "Constant Voltage Crossover Network Design", Richard Small, January, 1971; "Active and Passive Filters in Loudspeaker Crossover Networks", Ashley and Kaminsky, June 1971; and "A Novel Approach to Linear Phase Loudspeakers Using Passive Crossover Networks", E. Backgaard, May 1977.
When such prior art loudspeaker topologies are considered with reference to the poles and zeros of the resultant system input-output transfer function, it is found that all poles and zeros, that is all p-z, tend to disappear. In general most prior art loudspeaker designs or configurations have utilized acoustic summations which caused the disappearance of as many p-z as possible in the summation while tending toward some good approximation of the transfer function of Equation (1). The present invention takes an opposite approach, that is, retaining all, or as many as possible, of the p-z of the individual elements in the final summation. Also crucial to the method of this invention is the inclusion of transmission zeros in the design of the crossover filter circuits. These transmission zeros, taken together with retention in the total loudspeaker system of the dominant poles inherent in the separate crossover filter circuits allows the approximation of the transfer function of Equation (1) to a high degree of accuracy, while also overcoming the problems, mentioned above (1.1, 1.2 and 1.3), of the prior art loudspeaker designs.